"The overall law of proportionality" by Zeising
In the 19th century a large contribution to development of the theory of proportionality was made by the German scientist Zeising issued the book "Neue Lehre von den Proportionen des menschlichen Korpers" (1854). This one is until now by widely quoted book among the works dedicated to proportionality problem.
Outgoing from the fact that proportion is the ratio of two unequal parts between themselves and to the whole in their perfect combination Zeising formulates the law of proportionality as the following:
"The division of the whole on the unequal parts looked proportional when the ratio of parts of the whole between themselves is the same that the ratio of them to the whole, i.e. that, the ratio, which gives the golden section".
Attempting to prove that all Universe is subjected to this law Zeising tries to find it both in the organic and in the inorganic world.
To confirm this he gives diverse data about ratios of mutual distances between themselves of celestial heavenly bodies (corresponding to the golden section), also he finds the same ratio in the constitution of human body, in the configuration of minerals, in plants, in the sound chords of music, and in the architectural monuments.
By considering of Apollo and Venus statues Zeising finds that at division of the common altitude in the given ratio the line of division passes through natural partitionings of the human body. The first division passes through the navel, the second one through the middle of the neck etc., that is, all sizes of the separate body parts are obtained by the division of the whole in the golden section.
Analyzing a significance of the golden section law in music Zeising shows that the ancient Greeks assigned an aesthetic impression of chords to proportional division of the octave through the arithmetical and harmonic proportion.
Basing on the fact that only those combinations of tones are beautiful when they are in proportional ratio and that the combination of only two tones does not give a full harmony, Zeising shows that the most pleasant consonances have such combination of tones when the ratio of frequencies included in the chord is close to the golden proportion. For example, the combination of small third with the octave of the main tone corresponds to frequency ratio: 3:5; the combination of the large third with the octave of the main tone gives the frequency ratio: 5:8 (note that the numbers 3, 5, 8 are Fibonacci numbers!).
Further Zeising makes a conclusion that as these two-tone combinations among two-valued combinations are the most pleasant for hearing, it, apparently, explains that fact that only these combinations finish the musical periods. By using this fact Zeising explains why impromptu national melody and simple music of two French or English horns is gone in sixths and their supplements, the thirds.
Zeising pays attention for one curious fact. As is known, the major (man's) and minor (woman's) harmonies are constructed on the basis of the major and minor triad. The major triad constructed on the basis of the large third is a fine consonance since acoustical point of view. This one creates the impression of balance, physical perfection, light, vigor integrated in life by concept of "majority".
The minor triad constructed on the basis of the small third is a consonance, which is incorrect since acoustical point of view. This one creates the impression of the broken sounding and has a nature of gloominess, sadness, weakness integrated in life by concept of "minority".
In this connection Zeising notes that the combination of the octave with the large third of the main tone corresponds to the ratio of the lower and upper parts of man's body, and the combination of the octave with the small third of the main tone corresponds to the ratio of the lower and upper parts of woman's body.
Passing to a significance of the law of proportionality in architecture Zeisung shows that the architecture in the field of arts takes the same place as well as the organic world in the nature inspiring the inert matter on the basis of world's laws. Systematization, symmetry and proportionality thus are its indispensable attributes; it follows from here that the problem on the proportionality laws stands considerably more acute in architecture, than in sculpture or in painting.